Digital Signal Processing Reference
In-Depth Information
where
<
P
+
Q
P
−
Q
,
λ
=
λ
r
+
j
λ
i
,
α
=
α
r
+
j
α
i
,
κ
=
(37)
:
κ
(|
α
| −
α
i
)
2
|
α
| +
α
i
2
κ
µ
1
=
,
µ
2
=
sign
(
α
r
)
.
Using the parameters P, Q, and R, the controllability Gramian
K
(
b
)
0
and the observability
Gramian
W
(
b
)
0
of the balanced realization
(
A
b
,
B
b
,
C
b
, d
b
)
can be expressed as follows:
K
(
b
0
=
W
(
b
0
=
Θ
(38)
Θ
=
diag
(
θ
1
,
θ
2
)
P
2
−
Q
2
+
R,
P
2
−
Q
2
−
R
)
.
=
diag
(
(39)
3.3. Property of the positive definite symmetric matrix
P
In this subsection, we consider the property of the positive definite symmetric matrix
P
. The
following two theorems lead a symmetric property of the optimal positive definite symmetric
matrix
P
opt
[1].
Theorem 1.
[9] L
2
-sensitivity S
(
P
)
has the unique global minimum, which is achieved by a positive
definite symmetric matrix
P
opt
satisfying
∂
S
(
P
)
∂
P
=
0
.
(40)
P
=
P
opt
✷
Theorem 2.
[1] If a positive definite symmetric matrix
P
opt
satisfies
∂
S
(
P
)
∂
P
=
0
(41)
P
=
P
opt
then the positive definite symmetric matrix
ΣP
−
1
opt
Σ
also satisfies
∂
S
(
P
)
∂
P
=
0
(42)
P
=
ΣP
−
1
opt
Σ
for the signature matrix
Σ
which satisfies Eq. (29).
✷
